= Solution
Put $H_{12}=\operatorname{diag}(1,-1,0)$ and $H_{23}=\operatorname{diag}(0,1,-1)$. On the <root space> spanned by $E_{ij}$, their adjoint <eigenvalues> are $h_i-h_j$ and $k_i-k_j$; both act as zero on the <Cartan subalgebra>. Thus the <Killing form> is the sum of their products over the six <roots>. The pair of <roots> $\pm(L_1-L_2)$ contributes $-4$, the pair $\pm(L_2-L_3)$ contributes $-4$, and $\pm(L_1-L_3)$ contributes $2$. Hence
$$
\boxed{B(H_{12},H_{23})=-6.}
$$
This also agrees with the <Killing form of the special linear Lie algebra>, $B(A,B)=6\operatorname{tr}(AB)$ for $\mathfrak{sl}_3$.
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